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%I #18 Nov 04 2017 08:52:19
%S 2,-4,8,-14,18,-36,56,-74,116,-164,224,-324,442,-592,808,-1074,1410,
%T -1860,2416,-3102,4010,-5112,6464,-8204,10294,-12860,16072,-19914,
%U 24586,-30356,37248,-45534,55608,-67604,81928,-99182,119608,-143832,172760,-206834,247048,-294676,350504,-416080,493248,-583340,688616,-811740,954974,-1121564,1315504,-1540210,1800434,-2102060,2450224,-2852040
%N Row 2 in rectangular array A292929.
%H Paul D. Hanna, <a href="/A294065/b294065.txt">Table of n, a(n) for n = 0..380</a>
%e G.f.: A(q) = 2 - 4*q + 8*q^2 - 14*q^3 + 18*q^4 - 36*q^5 + 56*q^6 - 74*q^7 + 116*q^8 - 164*q^9 + 224*q^10 - 324*q^11 + 442*q^12 - 592*q^13 + 808*q^14 - 1074*q^15 + 1410*q^16 - 1860*q^17 + 2416*q^18 - 3102*q^19 + 4010*q^20 +...
%e RELATED SERIES.
%e Let R1(q) denote the g.f. of row 1 (with offset 0) in array A292929, then
%e A(q)/R1(q) = 1 + q^2 + q^3 - 3*q^5 + q^6 + 4*q^7 + q^8 - 3*q^9 + q^10 + 3*q^11 + q^12 - 5*q^13 + q^14 + 7*q^15 - 11*q^17 + 16*q^19 + 2*q^20 - 18*q^21 + 21*q^23 + q^24 - 27*q^25 + q^26 + 38*q^27 + q^28 - 55*q^29 + 2*q^30 +...
%e then it appears that the even bisection of A(q)/R1(q) forms a g.f. of A053692:
%e (A(q)/R1(q) + A(-q)/R1(-q))/2 = Product_{n>=1} (1 - q^(16*n))^2*(1 + q^(4*n-2)).
%t nmax = 55; kmax = Ceiling[Sqrt[nmax]];
%t Q[q_] := Sum[(x - q^k)^k, {k, -kmax, kmax}];
%t S[q_] := Sqrt[Q[q]/Q[-q]];
%t row[n_] := (1/q^n)*SeriesCoefficient[Sqrt[Q[q]/Q[-q]], {x, 0, n}] + O[q]^nmax // CoefficientList[#, q]&;
%t row[2] (* _Jean-François Alcover_, Nov 04 2017 *)
%Y Cf. A292929, A293132, A294066, A294067.
%K sign
%O 0,1
%A _Paul D. Hanna_, Oct 23 2017
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